Download Mean Semi-absolute Deviation Model for Uncertain Portfolio

Journal of Uncertain Systems
Vol.6, No.4, pp.299-307, 2012
Online at: www.jus.org.uk
Mean Semi-absolute Deviation Model for Uncertain Portfolio
Optimization Problem
Yixuan Liu1 , Zhongfeng Qin2,∗
1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2
School of Economics and Management, Beihang University, Beijing 100191, China
Received 10 June 2012; Revised 1 October 2012
Abstract
Semi-absolute deviation is a commonly used downside risk measure in the portfolio optimization problem. However, there is no literature on taking semi-absolute deviation as a risk measure in the framework
of uncertainty theory. This paper fills the gap by means of defining semi-absolute deviation for uncertain
variables and establishes the corresponding mean semi-absolute deviation models in uncertain environment. Finally, numerical examples are presented to illustrate advantages of the proposed approach.
c
⃝2012
World Academic Press, UK. All rights reserved.
Keywords: uncertain variable, uncertainty measure, uncertain portfolio selection, semi-absolute deviation
1
Introduction
Portfolio optimization problem concerns with an individual who is trying to allocate one’s capital to a selected
number of securities in order to achieve the investment goal. The first mathematical model was proposed by
Markowitz [15], in which expected value was used to describe return and variance was used to measure risk.
Since then, variance has been widely used as a risk measure, and a large number of models (e.g. [2, 6]) have
been investigated. Different from the quadratic Markowitz model, a linear model for portfolio optimization
was provided by Konno and Yamazaki [10] in which the absolute deviation was used to measure the risk of
the portfolio. In particular, when the returns of securities are multivariate-normally distributed, the model
is equivalent to Markowitz’s mean-variance model. Based on absolute deviation, numerous models were
developed such as [1, 3, 23].
Although variance and absolute deviation are appropriate risk measures in many cases, they make no
distinction between gains and losses. In other words, they regard high returns that investors like as equally
undesirable as low returns that investors dislike. In order to solve the problem, downside risk measures were
proposed in [4, 12, 15, 22, 24] and proved as reasonable measures [21, 26] since they only take the negative
deviations from the expected level into account. Thus, downside risk measures would help investors make
proper investments when the returns are asymmetrical. One of the commonly used downside risk measures is
semivariance originally introduced in [15] and developed by several scholars such as [5, 16]. The other wellknown downside risk measure is semi-absolute deviation proposed by Speranza [24]. In contrast to Markowitz’s
mean-semivariance model, the mean semi-absolute deviation model for portfolio optimization can be easily
evaluated since it can be determined using linear programming models [17, 23, 24].
In the above works, security returns are considered as random variables. However, it is difficult to use
probability theory to model the problem in some situations such as the emerging market. In order to deal
with vagueness and ambiguity of the returns on securities, fuzzy set theory may be used to describe the fuzzy
behavior in portfolio optimization problem by means of fuzzy variable. In fact, fuzz portfolio optimization
has been widely investigated in different methods: fuzzy goal programming, fuzzy compromise programming,
fuzzy decision theory, possibilistic programming [25], interval programming [11] and other fuzzy formulations
[27]. The detailed exposition may be found in [28] which gave an overview on the development of fuzzy
∗ Corresponding
author. Email: [email protected] (Z. Qin).
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Y. Liu and Z. Qin: Mean Semi-absolute Deviation Model for Uncertain Portfolio Optimization Problem
portfolio optimization. Different from the above approaches, fuzzy portfolio optimization was also considered
by Huang [7] and Qin et. al [18] based on credibility measure.
With the development of the theory of dealing with subjective uncertainty, Liu [13] found a paradox when
fuzzy variables are employed to describe subjective uncertain phenomena. In order to overcome this shortcoming, Liu [13] proposed uncertain measure and further found uncertainty theory. Surrounding the subject, some
works have been done. In particular, several researchers have considered the portfolio optimization problem in
which security returns are assumed to be uncertain variables. Qin et al. [20] introduced mean-variance model
for uncertain portfolio optimization. Huang [8] proposed mean-risk model for the problem by introducing a
risk curve. In addition, Huang [9] also established risk index based models for uncertain portfolio adjusting
problem.
For the analogous reasons mentioned in stochastic environment, variance and absolute deviation become
unreasonable in the case that uncertain returns are asymmetrical. Therefore, we have to consider downside
risk measures for uncertain returns. As a popular downside risk measure, semi-absolute deviation is a direct
and comparatively simple approach. However, semi-absolute deviation has not been studied in the framework
of uncertainty theory. Thus, this paper will define semi-absolute deviation for uncertain variables and establish
new optimization models for uncertain portfolio optimization problem.
The rest of the paper is organized as follows. In Section 2, we recall some basic concepts in uncertainty
theory. In Section 3, we define semi-absolute deviation for uncertain variables and discuss the mathematical
properties. In Section 4, we present three mean semi-absolute deviation models for uncertain portfolio optimization by optimizing different objectives. Section 5 presents a numerical example to illustrate advantages
of the proposed approach. Further, some conclusions are given in Section 6.
2
Preliminaries
In 2007, Liu [13] proposed the concept of uncertain measure and founded uncertainty theory. In this part, we
recall some basic definitions and properties about uncertain measure and uncertain variables, which will be
used in the whole paper.
Let Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ ∈ L is called an event. It
is necessary to assign to each event Λ a number M{Λ} which indicates the chance that Λ will occur. Liu
[13] proposed the following four axioms to ensure that the number M{Λ} satisfying certain mathematical
properties,
Axiom 1. (Normality) M{Γ} = 1.
Axiom 2. (Monotonicity) M{Λ1 } ≤ M{Λ2 } whenever Λ1 ⊂ Λ2 .
Axiom 3. (Self-Duality) M{Λ} + M{Λc } = 1 for any event Λ.
Axiom 4. (Countable Subadditivity) For every countable sequence of events {Λi }, we have
M
{∞
∪
i=1
}
Λi
≤
∞
∑
M{Λi }.
(1)
i=1
The triplet (Γ, L, M) is called an uncertainty space. An uncertain variable ξ is defined by Liu [13] as a
measurable function from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of
real numbers, the set {ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B} is an event. An uncertain variable ξ can be characterized
by an uncertainty distribution which is a function Φ : R → [0, 1] defined by Liu [13] as Φ(t) = M{ξ ≤ t}.
For example, by a linear uncertain variable, we mean that the variable has the following linear uncertainty
distribution

0,
if r ≤ a,


(r − a)/(b − a), if a ≤ r ≤ b,
Φ(r) =


1,
if r ≥ b.
The linear uncertain variable is denoted by L(a, b) where a and b are real numbers with a < b. By a zigzag
301
Journal of Uncertain Systems, Vol.6, No.4, pp.299-307, 2012
uncertain variable, we mean that the variable has the following

0,




(r − a)/2(b − a),
Φ(r) =

(x + c − 2b)/2(c − b),



1,
zigzag uncertainty distribution
if
if
if
if
r ≤ a,
a ≤ r ≤ b,
b ≤ r ≤ c,
r ≥ c.
The zigzag uncertain variable is denoted by Z(a, b, c) where a, b, c are real numbers with a < b < c. By a
normal uncertain variable, we mean that the variable has the following normal uncertainty distribution
(
(
))−1
π(e − r)
√
Φ(r) = 1 + exp
, r ∈ R.
3σ
The normal uncertain variable is denoted by N (e, σ) where e and σ are real numbers with σ > 0.
The uncertain variables ξ1 , ξ2 , . . . , ξm are said by Liu [13] to be independent if
{m
}
∩
M
{ξi ∈ Bi } = min M {ξi ∈ Bi }
1≤i≤m
i=1
(2)
for any Borel sets B1 , B2 , . . . , Bm of real numbers. In order to rank uncertain variables, the expected value
of ξ was proposed by Liu [13] as
∫ +∞
∫ 0
E[ξ] =
M{ξ ≥ r}dr −
M{ξ ≤ r}dr
(3)
−∞
0
provided that at least one of the two integrals is finite. For example, the linear uncertain variable ξ ∼ L(a, b)
has an expected value E[ξ] = (a + b)/4; the zigzag uncertain variable ξ ∼ Z(a, b, c) has an expected value
E[ξ] = (a + 2b + c)/4; the normal uncertain variable ξ ∼ N (e, σ) has an expected value e, i.e., E[ξ] = e.
Further, if fuzzy variables ξ and η are independent, then
E[aξ + bη] = aE[ξ] + bE[η].
(4)
for any a, b ∈ ℜ. In particular, we have E[aξ + b] = aE[ξ] + b.
3
Semi-absolute Deviation of Uncertain Variables
In this section, we give the definition of the semi-absolute deviation of uncertain variables, and prove some
mathematical properties for it. Let ξ be an uncertain variable and e a real number. For simplicity, we write
(ξ − e)− = min(ξ − e, 0) and (ξ − e)+ = max(ξ − e, 0).
Definition 1 Let ξ be an uncertain variable with finite expected value e. Then the semi-absolute deviation of
ξ is defined as
[
]
Sa[ξ] = E |(ξ − e)− | .
(5)
Remark 1: Definition 5 tells us that the semi-absolute deviation is the expected value of |(ξ − e)− |. Since
|(ξ − e)− | is a nonnegative uncertain variable, we also have
∫ ∞
{
}
Sa[ξ] =
M |(ξ − e)− | ≥ r dr
∫0 ∞
=
M {e − ξ ≥ r} dr
∫0 e
=
M {ξ ≤ r} dr
−∞
∫ e
=
Φ(r)dr
(6)
−∞
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Y. Liu and Z. Qin: Mean Semi-absolute Deviation Model for Uncertain Portfolio Optimization Problem
where Φ(·) is the uncertainty distribution of ξ. This formula will facilitate the calculation of semi-absolute
deviation in many cases.
Example 1: Let ξ ∼ L(a, b) be a linear uncertain variable. Then its semi-absolute deviation is
∫
∫
E[ξ]
Sa[ξ] =
(a+b)/2
Φ(r)dr =
−∞
a
r−a
1
dr =
b−a
b−a
∫
(a+b)/2
(r − a)dr =
a
(b − a)2
.
8
Example 2: Let ξ ∼ Z(a, b, c) be a zigzag uncertain variable. The semi-absolute deviation of ξ is

(3(b − a) + (c − b))2


,

(a+2b+c)/4
64(b − a)
Sa[ξ] =
Φ(r)dr =

((b − a) + 3(c − b))2
−∞


,
64(c − b)
∫
if b − a ≥ c − b,
if b − a ≤ c − b,
which has alternative expression
Sa[ξ] =
(2c − 2a + |2b − a − c|)2
.
32(c − a + |2b − a − c|)
Example 3: Let ξ ∼ N (e, σ) be a normal uncertain variable. Then the semi-absolute deviation of ξ is
∫
∫
e
Sa[ξ] =
e
Φ(r)dr =
−∞
0
√
(
(
))−1
3σ ln 2
π(e − r)
√
1 + exp
dr =
.
π
3σ
Theorem 1 Let ξ be an uncertain variable with a finite expected value. Then for any real numbers a and b,
we have
Sa[aξ + b] = |a| · Sa[ξ].
(7)
Proof: It follows from the definition of semi-absolute deviation that
Sa[aξ + b] = E[|(aξ + b − aE[ξ] − b)− |] = E[|a(ξ − E[ξ])− |] = |a| · E[|(ξ − E[ξ])− |] = |a| · Sa[ξ].
The proof is complete.
Theorem 2 Let ξ be an uncertain variable with expected value e, and let Sa[ξ] be the semi-absolute deviation
of ξ. Then we have 0 ≤ Sa[ξ] ≤ E[|ξ − e|].
Proof: Since Sa[ξ] is nonnegative, for any real number r, we have
{γ | |ξ(γ) − e| ≥ r} ⊃ {γ | |(ξ(γ) − e)− | ≥ r}.
It follows from the monotonicity of uncertain measure that
M{|ξ − e| ≥ r} ≥ M{|(ξ − e)− | ≥ r}, ∀ r.
It immediately follows from the definition of semi-absolute deviation that
∫
+∞
−
∫
+∞
M{|(ξ − e) | ≥ r}dr ≤
Sa[ξ] =
0
M{|ξ − e| ≥ r}dr = E[|ξ − e|].
0
The proof is complete.
Theorem 3 Let ξ be an uncertain variable with finite expected value e. Then Sa[ξ] = 0 if and only if
M{ξ = e} = 1.
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Journal of Uncertain Systems, Vol.6, No.4, pp.299-307, 2012
Proof: If Sa[ξ] = 0, then E[|(ξ − e)− |] = 0. By the definition of expected value, we have
−
∫
+∞
E[|(ξ − e) |] =
M{|(ξ − e)− | ≥ r}dr,
0
which implies M{|(ξ − e)− | ≥ r} = 0 for any r > 0. Further, it is obtained that
M{|(ξ − e)− | = 0} = 1,
(8)
which implies ξ − e = (ξ − e)+ + (ξ − e)− = (ξ − e)+ almost everywhere. Thus, we have
∫
+∞
M{(ξ − e)+ ≥ r}dr = E[(ξ − e)+ ] = E[ξ − e] = 0.
0
This indicates that M{(ξ − e)+ ≥ r} = 0 for any r > 0. Considering the self-duality of uncertainty measure,
we have
M{(ξ − e)+ = 0} = 1.
(9)
If follows from equations (8) and (9) that M{ξ − e = 0} = 1, i.e., M{ξ = e} = 1. Conversely, if M{ξ = e} = 1,
then we have M{|ξ − e| = 0} = 1 and M{|ξ − e| ≥ r} = 0 for any real number r > 0, which implies that
E[|ξ − e|] = 0. Further, it follows from Theorem 2 that Sa[ξ] = 0. The proof is complete.
4
Mean Semi-absolute Deviation Models
In this section, we establish new portfolio optimization models in uncertain environment by employing the
semi-absolute deviation of the portfolio as the measure of risk. Let ξi (i = 1, 2, . . . , n) be the uncertain return
of the ith security, and xi be the proportion of total amount of funds invested in security i. By uncertain
arithmetic, the total return of the portfolio is ξ1 x1 + ξ2 x2 + · · · + ξn xn which is also an uncertain variable.
This means that the portfolio is risky. If an investor wants to maximize the expected return at the given risk
level, then it can express in a single-objective non-linear programming model as follows,

maximize
E[ξ1 x1 + ξ2 x2 + · · · + ξn xn ]



 subject to: Sa[ξ x + ξ x + · · · + ξ x ] ≤ d,
1 1
2 2
n n
(10)

x
+
x
+
·
·
·
+
x
=
1,
1
2
n



xi ≥ 0, i = 1, 2, . . . , n,
where d denotes the maximum risk level the investors can tolerate. The constraints ensure that all the capital
will be invested to n securities and short sales are not allowed. It is worth pointing out that the proposed
model mainly deal with the case with asymmetrical return.
When an investor wants to minimize the risk of investment with an acceptable expected return level, the
mathematical formulation of the problem is given as follows,

minimize
Sa[ξ1 x1 + ξ2 x2 + · · · + ξn xn ]



 subject to: E[ξ x + ξ x + · · · + ξ x ] ≥ r,
1 1
2 2
n n
(11)

x1 + x2 + · · · + xn = 1,



xi ≥ 0, i = 1, 2, . . . , n,
where r is the minimum expected return level accepted by the investor.
A risk-averse investor always wants to maximize the return and minimize the risk of the portfolio. However,
these two objects are inconsistent. To determine an optimal portfolio with a given degree of risk aversion, we
formalize the following optimization model,

E[ξ1 x1 + ξ2 x2 + · · · + ξn xn ] − ϕ · Sa[ξ1 x1 + ξ2 x2 + · · · + ξn xn ]
 maximize
subject to: x1 + x2 + · · · + xn = 1,
(12)

xi ≥ 0, i = 1, 2, . . . , n,
304
Y. Liu and Z. Qin: Mean Semi-absolute Deviation Model for Uncertain Portfolio Optimization Problem
where ϕ ∈ [0, +∞) represents the degree of absolute risk aversion. Here, the greater the value of ϕ is, the
more risk-averse the investors are. Note that ϕ = 0 means that the investor does not consider risk, and ϕ
approaching infinity means that the investor will allocate all the money to risk-less securities.
Note that in uncertain environment, E[ξ1 x1 + ξ2 x2 · · · + ξn xn ] ̸= x1 E[ξ1 ] + x2 E[ξ2 ] + · · · + xn E[ξn ] for
general uncertain variables ξ1 , ξ2 , . . . , ξn . However, the inequality will become equality when ξ1 , ξ2 , . . . , ξn
are independent. In particular, if security returns are all linear uncertain variables, denoting∑the return
n
of∑security i∑by ξi = (ai , bi ). It follows from Theorem 1.21 of [14] that the portfolio return i=1 ξi xi =
n
n
( i=1 xi ai , i=1 xi bi ) is also a linear uncertain variable. Based on this, model (11) may be converted into
the following a quadratic programming,

( n
)2

∑



minimize
xi (bi − ai )




i=1

n
∑
(13)
subject to:
xi (ai + bi ) ≥ 2r,



i=1




x1 + x2 + · · · + xn = 1,


xi ≥ 0, i = 1, 2, . . . , n.
Further, we assume that security returns are all zigzag uncertain variables, denoting the return
∑n of secuthat
the
portfolio
return
rity
i
by
ξ
=
(a
,
b
,
c
).
It
follows
from
Theorem
1.22
of
[14])
i
i
i
i
i=1 ξi xi =
∑n
∑n
∑n
( i=1 xi ai , i=1 xi bi , i=1 xi ci ) is also a zigzag uncertain variable. Then the model (11) may be converted
into the following deterministic form,
n
)2

( n
∑
∑


2xi (ci − ai ) + xi (2bi − ai − ci )



i=1

i=1

 minimize
n
n
∑

∑


x
(c
−
a
)
+
x
(2b
−
a
−
c
)

i i
i
i
i
i
i i=1
i=1
(14)
n
∑


 subject to:
x
(a
+
2b
+
c
)
≥
4r,

i
i
i
i



i=1



x1 + x2 + · · · + xn = 1,


xi ≥ 0, i = 1, 2, . . . , n.
Remark 2: When the security returns are all linear or zigzag uncertain variables, both models (10) and (12)
can be converted into deterministic programming problems in a similar way.
5
A Numerical Example
In this section, we present a numerical example about uncertain portfolio optimization problem to demonstrate
the new modeling idea of this paper. For this purpose, we consider the problem that an investor plans to invest
his fund among twenty securities. It is worth pointing out that the experiment process may be repeated for
more and/or less securities. Further, all the future returns of securities are zigzag uncertain variables denoted
by ξi = Z(ai , bi , ci ) for i = 1, 2, . . . , 20. For the practical investment problem, the key is to accurately estimate
the values of parameters ai , bi and ci for i = 1, 2, . . . , 20. The detailed method may be found in the chapter
of Uncertain Statistics in [14]. Once these parameters are obtained, we can apply the proposed model to
construct an optimal portfolio for the investor according to his/her requirements. The assumed returns of
securities are shown in Table 1.
Since all the security returns are zigzag uncertain variables, the proposed mean semi-absolute deviation
models can be converted into equivalent deterministic models. Here, we use the model (11) to determine the
optimal portfolio. The transformed deterministic model is as follows,

20
)2 /( 20
20
)
( 20

∑
∑
∑
∑


2xi αi + xi βi xi αi + xi βi  minimize


i=1
i=1
i=1
i=1

(15)
subject
to:
x
µ
+
x
µ
+
·
·
·
+
x
µ
≥
r,
1 1
2 2
20 20




x1 + x2 + · · · + x20 = 1,



0 ≤ xi ≤ 0.5, i = 1, 2, . . . , 20
305
Journal of Uncertain Systems, Vol.6, No.4, pp.299-307, 2012
Table 1: Zigzag uncertain returns of 20 securities
Security No.
1
2
3
4
5
6
7
8
9
10
Uncertain Return
(−0.12, 0.05, 0.21)
(−0.19, 0.02, 0.22)
(−0.11, −0.01, 0.13)
(−0.17, 0.02, 0.16)
(−0.19, 0.03, 0.19)
(−0.13, 0.01, 0.20)
(−0.18, 0.01, 0.25)
(−0.15, 0.05, 0.18)
(−0.21, −0.01, 0.18)
(−0.09, 0.02, 0.13)
Security No.
11
12
13
14
15
16
17
18
19
20
Uncertain Return
(−0.16, −0.01, 0.14)
(−0.13, 0.01, 0.19)
(−0.14, 0.03, 0.21)
(−0.18, 0.02, 0.16)
(−0.14, 0.00, 0.18)
(−0.17, 0.00, 0.33)
(−0.24, 0.02, 0.45)
(−0.14, 0.03, 0.21)
(−0.12, 0.02, 0.14)
(−0.20, 0.01, 0.24)
where αi = ci − ai , βi = 2bi − ai − ci and µi = (ai + 2bi + ci )/4. We employ fmincon in MATLAB 7.1 to solve
the above model. For given minimal return level r, we obtain a series of optimal investment strategies. The
computational results are shown in Table 2 in which the first column is the preset minimal return level and
the second column is semi-absolute deviation of the optimal portfolio. Further, the efficient frontier of model
(15) are shown in Figure 1.
Table 2: Investment proportion to 20 securities with zigzag uncertain return rates (%)
r
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
6
Semi-Absolute Deviation
0.922
0.922
0.947
0.982
1.025
1.068
1.145
1.324
1.560
1.803
2.051
( x1 ,
( 0.00,
( 0.00,
( 0.00,
( 7.69,
(23.08,
(38.46,
(50.00,
(50.00,
(50.00,
(50.00,
(50.00,
x3 ,
50.00,
50.00,
16.67,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
x8 , x10 ,
0.00, 50.00,
0.00, 50.00,
0.00, 50.00,
0.00, 50.00,
0.00, 50.00,
0.00, 50.00,
10.00, 40.00,
50.00, 0.00,
33.33, 0.00,
16.67, 0.00,
0.00, 0.00,
x17 , x19 )
0.00, 0.00)
0.00, 0.00)
0.00, 33.33)
0.00, 42.31)
0.00, 26.92)
0.00, 11.54)
0.00, 0.00)
0.00, 0.00)
16.67, 0.00)
33.33, 0.00)
50.00, 0.00)
Conclusion
In this paper, the concept of semi-absolute deviation was first defined for uncertain variables, and some
properties were proven. By means of taking semi-absolute deviation as a risk measure, three mean semiabsolute deviation models were presented in this work. A numerical example was presented and solved in the
situation that all the security returns are zigzag uncertain variables. The computational results implied that
the proposed model is feasible and significative for uncertain portfolio optimization problem.
Acknowledgments
This work was supported by National Natural Science Foundation of China Grant No.71001003 and Specialized
Research Fund for the Doctoral Program of Higher Education of China Grant No. 20101102120022.
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Y. Liu and Z. Qin: Mean Semi-absolute Deviation Model for Uncertain Portfolio Optimization Problem
0.055
0.05
0.045
Expected Return
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
1
1.5
2
Semi-Absolute Deviation (Risk)
2.5
Figure 1: Efficient frontier of mean semi-absolute deviation model (15)
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